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Help: analytical of 2nd order PDE

  1. Oct 29, 2011 #1
    What method can I use to analytically solve the following 2nd order PDE?
    u=u(x,t)
    ∂u/∂t - a*x*∂u/∂x-D*∂^{2}u/∂t^{2} = 0
    I.C.: u(x,t=0)=u_i
    B.C.: u(x=+∞)=0
    u(x=-∞)=1

    Is self-similar the only way to solve it, or is there any other method can be used to solve it?
    How to set the similarity variable if I use self-similar method?

    Thanks
     
  2. jcsd
  3. Oct 30, 2011 #2
    I use separation variable:
    u(x,t)=F(η)T(t): η=sqrt(a/D) x
    Then obtain the following 2 ODEs:
    (1)T'+a λ T=0
    (2)F"+η F'+λ F=0

    How to solve the second ODE?
     
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